Transition Pathways in Complex Systems: Application of the Finite-Temperature String Method to the Alanine Dipeptide

Size: px
Start display at page:

Download "Transition Pathways in Complex Systems: Application of the Finite-Temperature String Method to the Alanine Dipeptide"

Transcription

1 Transition Pathways in Complex Systems: Application of the Finite-Temperature String Method to the Alanine Dipeptide Weiqing Ren Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Eric Vanden-Eijnden Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA Paul Maragakis Department of Chemistry and Chemical Biology, Harvard University, Cambridge MA, USA and Laboratoire de Chimie Biophysique, Institut de Science et d Ingènierie Supramoléculaires, Université Louis Pasteur, 8 rue Gaspard Monge, F Strasbourg, France Weinan E Department of Mathematics and PACM, Princeton University, Princeton, NJ 08544, USA (Dated: July 14, 2005) 1

2 Abstract The finite-temperature string method proposed by E, Ren and Vanden-Eijnden is a very effective way of identifying transition mechanisms and transition rates between metastable states in systems with complex energy landscapes. In this paper, we discuss the theoretical background and algorithmic details of the finite-temperature string method, as well as application to the study of isomerization reaction of the alanine dipeptide, both in vacuum and in explicit solvent. We demonstrate that the method allows us to identify directly the iso-committor surfaces, which are approximated by hyperplanes, in the region of configuration space where the most probable transition trajectories are concentrated. These results are verified subsequently by computing directly the committor distribution on the hyperplanes that define the transition state region. PACS numbers: r, x, Pm 2

3 I. INTRODUCTION This paper is a continuation of the works presented in Refs. 1 and 2 on the study of transition pathways and transition rates between metastable states in complex systems. In the present paper, we will focus on the finite-temperature string method (FTS), presented in Ref. 2. We will discuss the theoretical background behind FTS, the algorithmic details and implementation issues, and we will demonstrate the success and difficulties of FTS for the example of conformation changes of an alanine dipeptide molecule, both in vacuum and with explicit solvent. The study of transition between metastable states has been a topic of great interest in many areas for many years. For systems with simple energy landscapes in which the metastable states are separated by a few isolated barriers, there exists a solid theoretical foundation as well as satisfactory computational tools for identifying the transition pathways and transition rates. The key objects in this case are the transition states, which are saddle points on the potential energy landscape that separate the metastable states. The relevant notion for the transition pathways is that of minimum energy paths (MEP). MEPs are paths in configuration space that connects the metastable states along which the potential force is parallel to the tangent vector. MEP allows us to identify the relevant saddle points which act as bottlenecks for a particular transition. Several computational methods have been developed for finding the MEPs. Most successful among these methods are the nudged elastic band (NEB) method 3 and the zero-temperature string method (ZTS). 1 Once the MEP is obtained, transition rates can be computed using several strategies (see for example Refs. 1 and 4). Alternatively, one may look directly for the saddle point, and several methods have been developed for this purpose, including the conjugate peak refinement method, 18 the ridge method 19 and the dimer method. 20 The situation is quite different for systems with rough energy landscapes, as is the case for typical chemical reactions of solvated systems. In this case, traditional notions of transition states have to be reconsidered since there may not exist specific microscopic configurations that identify the bottleneck of the transition. Instead the potential energy landscape typically contains numerous saddle points, most of which are separated by barriers that are less than or comparable to k B T, and therefore do not act as barriers. There is not a unique most probable path for the transition. Instead, a collection of paths are important. 21 Describing 3

4 transitions in such systems has become a central issue in recent years. The standard practice for identifying transition pathways and transition rates in complex systems is to coarse-grain the system using a pre-determined reaction coordinate, usually selected on an empirical basis. The free energy landscape associated with this reaction coordinate is then calculated, using for example the blue-sampling technique, 5 and the transition states are identified as the saddle points in the free energy landscape. Such a procedure based on intuitive notions of reaction coordinates has been both helpful and misleading. For simple enough systems, for which a lot is known about the mechanisms of the transition, this procedure allows one to make use of and further refine the existing knowledge. But as has been pointed out by Chandler and collaborators, 6,7 if little is known about the mechanism of the transition, a poor choice of the reaction coordinate can lead to wrong predictions for the transition mechanism. Furthermore, intuitively reasonable coarsegrained variables may not be good reaction coordinates. 7 In view of this, it is helpful to make the notion of reaction coordinates more precise in order to place our discussions on a firm basis. There is indeed a distinguished reaction coordinate, defined with the help of the backward Kolmogorov equation, which specifies, at each point of the configuration space or phase space, the probability that the reaction will succeed if the system is initiated at that particular configuration or phase space location (see the discussion in section II). All other reaction coordinates are approximate, and the quality of the approximation can in principle be quantified. This particular notion of reaction coordinate is crucial for our discussion, since it identifies the so-called iso-committor surfaces, which is a central object in our study. There are several ways of characterizing this distinguished reaction coordinate that identifies the iso-committor surfaces. Besides being the solution of the backward Kolmogorov equation, it also satisfies a variational principle, as we discuss below. Nevertheless, it is often impractical to find this reaction coordinate, since it is an object that may depend on very many variables. The backward Kolmogorov equation is an equation in a huge dimensional space. What we are really interested in, after all, are the ensemble of transition trajectories. This point of view has been emphasized by Chandler and co-workers, in their development of the transition path sampling technique (TPS). 6,7 Our viewpoint is consistent with that of TPS, but there is a crucial difference: TPS views the transition path ensemble in the space of trajectories parameterized by the physical time, 4

5 whereas we view the transition path ensemble in the configuration space with parameterization chosen for the convenience of computations. As we show below, this latter viewpoint offers a number of advantages, the most important of which is that in configuration space, the transition path ensemble can be identified by looking at the equilibrium distribution of the system restricted on the family of the iso-committor surfaces that separate the metastable sets. We assume that the transition path ensemble is localized in the configuration space, i.e. they form isolated tubes. We call them transition tubes. Our primary objective is to identify the transition tubes as well as the iso-committor surfaces within the tubes. By focusing on the transition tubes, we reduce a problem in large dimensional space, the backward Kolmogorov equation, to a large coupled system in one dimensional space, along the tube. With this understanding, we can easily appreciate the basic principles behind the finitetemperature string method. Consistent with the localization assumption, we approximate the iso-committor surfaces within the tubes by hyperplanes. We then adopt the variational characterization of the iso-committor surfaces (see (10)), but we restrict the trial functions to a family of hyperplanes parametrized by a curve that passes through the center of mass (with respect to the equilibrium distribution) on each plane. We call this curve the string. The finite-temperature string method is a way of finding the family of minimizing hyperplanes. It is worth noting that in the case when the energy landscape is simple and smooth, the transition tube reduces to MEP in the zero-temperature limit. At the same time, the finite-temperature string method reduces to the zero-temperature string method. 1 This paper is organized as follows. In section II, we recall some theoretical background for describing transition in complex systems. In section III we discuss the finite-temperature string method, its foundation and implementation. FTS is then used in section IV to study the conformation changes of the alanine dipeptide, first in vacuum (section IV A) and then in explicit solvent (section IV B). In each case we identify the transition tube using FTS, analyze the reaction coordinates and transition states. This is possible since FTS gives the iso-committor surfaces near the transition tube. 5

6 II. THEORETICAL BACKGROUND We begin with some theoretical background. To simplify the presentation, we will focus on the case when the system of interest obeys over-damped friction dynamics: γẋ = V (x) + 2γβ 1 η (1) where V is the potential energy of the system, γ is the friction coefficient, η is a white noise forcing, and β = 1/k B T is the inverse temperature. More general forms of Langevin dynamics are considered in Ref. 9, where it is shown that the mechanism of transition is unaffected by the value of the friction coefficient at least under the assumption that the mechanism of transition can be described within the configuration space alone. The system described by (1) has a standard equilibrium distribution ρ e (x) = Z 1 e βv (x) (2) in the configuration space Ω, where Z = Ω e βv (x) dx is a normalization constant. We will assume that there are two metastable states described by two subsets, A and B, of the configuration space Ω respectively, and we will refer to A and B as being the metastable sets. For simplicity, we will assume that there are no other metastable sets, i.e. ρ e (x)dx 1 (3) A B We are interested in characterizing the mechanism of transition between the metastable sets A and B. In the simplest situation when the energy landscape is smooth and the metastable states are separated by a few isolated barriers, this is usually done by identifying the transition states, which are saddle points on the potential energy landscape. The most probable path for the transition is the so-called minimum energy path which is the path in the configuration space such that the potential force is parallel to the tangent along the path. 1,3 As has been discussed earlier, 2,6 these concepts are no longer appropriate if the energy landscape is rough with many saddle points, most of which are separated by potential energy barriers that are less than or comparable to k B T, and therefore do not act as barriers for the transition. In this case, the notion of transition states has to be replaced by a transition state ensemble which is a probability distribution of states. 8,9 6

7 A. Transition path ensemble in trajectory space How do we characterize such an ensemble and the most probable transition paths? To address this question, Chandler et al proposed studying the ensemble of reactive trajectories, which are trajectories that successfully make the transition, parametrized by the physical time. By assigning appropriate probability densities to these trajectories, Monte Carlo procedures can be developed to sample the reactive trajectories. Indeed this is the basic idea behind transition path sampling 6,7 (see also Ref. 10 for an alternative technique to sample reactive trajectories). The efficiency gain of TPS over standard molecular dynamics comes from the fact that reactive trajectories are much shorter than the time it takes between successive transitions. Therefore, TPS allows one to sample much more reactive trajectories than in a standard molecular dynamics run. Each reactive trajectory can be post-processed to determine the committor value of the points along this trajectory, i.e. the probability that the reaction will succeed if initiated at that particular point. The collection of points with committor value close to 1 2 form the transition state ensemble. TPS in principle gives a systematic procedure for studying reactive paths in complex systems. In practice, however, it has several difficulties. First of all, harvesting enough reactive paths for statistical analysis might be very demanding. This task is not made easier by the fact the paths are parametrized by physical time, and these trajectories can be quite long compared to the time-step used in the simulations. In addition, the ensemble of reactive trajectories does not give directly the main object of interest, the transition state ensemble, and it requires a very non-trivial post-processing step to analyze the reactive paths in order to extract the transition state ensemble. As explained before, this post-processing step involves launching trajectories at each point along each reactive path to determine the committor value of that point, and this can be quite tedious indeed. 11 B. Transition path ensemble in configuration space An alternative viewpoint was proposed in Ref. 2 in connection with the finite-temperature string method. Its theoretical background was explained in Refs. 8 and 9. Instead of considering the ensemble of dynamic trajectories parametrized by the physical time, one views the transition paths in the configuration space. To understand why it is advantageous 7

8 to do so, it is helpful to consider first a distinguished reaction coordinate q(x) defined by the solutions of the backward Kolmogorov equation: V q + β 1 q = 0, q x A = 0, q x B = 1, (4) The solution to this equation has a very simple probabilistic interpretation: 12 q(x) is the probability that a trajectory initiated at x reaches the metastable set B before it reaches the set A. In other words, in terms of the reaction from A to B, q(x) gives the probability that this reaction will succeed if it has already made it to the point x. The level sets (or iso-surfaces) of q, Γ z = {x : q(x) = z}, z [0, 1], define the isocommittor surfaces: For any fixed z, trajectories initiated at points on Γ z have the same probability to reach B before reaching A. The iso-committor surfaces allow one to define the transition path ensemble in the configuration space in a different way, by restricting the equilibrium distribution (2) on these surfaces (see figure 1). To see how this is done, consider an arbitrary surface S in the configuration space and an arbitrary point x on S. The probability density of hitting S at x by any typical trajectory is ρ S (x) = Z 1 S e βv (x) (5) where the normalization factor Z S is the following integral over the surface S: Z S = S e βv (x) dσ(x), where dσ(x) is the surface element on S. (Note that ρ S (x) is a density that is attached to and lives in S, and that is why Z S is not equal to the normalization factor Z in (2)). On the other hand, if we consider only reactive trajectories, then this probability density changes to ρ S (x) = Z 1 S q(x)(1 q(x))e βv, (6) where Z S = q(x)(1 S q(x))e βv dσ(x). This can be seen as follows (see also Ref. 9). ρ S (x) is equal to the probability density that a trajectory (whether reactive or not) hits S at x, times the probability that the trajectory came from A in the past and the probability that it reaches B before reaching A in the future. Using the strong Markov property and statistical time reversibility, this latter quantity is given by q(x)(1 q(x)), and this gives (6). If S is an iso-committor surface, we see that ρ S = ρ S, i.e., the probability of hitting an iso-committor surface at x by a hopping trajectory is simply the equilibrium distribution restricted on the surface. 8

9 Let ρ z (x) = ρ S (x) when S = Γ z, the iso-committor surfaces labeled by z [0, 1]. We can now define the transition path ensemble as the one-parameter family of probability densities {ρ z (x), z [0, 1]} on the iso-committor surfaces. The discussion above tells us that ρ z (x) gives the target distribution when reactive trajectories hit the surface Γ z. C. Transition tubes in configuration space In principle the set on which the transition path ensemble concentrates can be a very complicated set. In the following, we will assume the distribution ρ z (x) is singly peaked on each iso-committor surface, and consequently the set becomes a localized tube. The case when ρ z is peaked at several isolated locations and hence there are several isolated tubes is similar. The tube can be characterized by its centerline and width. To be more precise, the centerline is defined as ϕ(z) = x Γz (7) where the average is with respect to ρ z (x) which is the restricted equilibrium distribution on Γ z. The width of the tube can be characterized by the eigenvalues of the covariance matrix C(z) = (x ϕ(z)) (x ϕ(z)) Γz. (8) The tube can be parameterized by other parameters, e.g. the normalized arc-length of the centerline. Remark. The transition tube can also be characterized by the probability that the reactive trajectories stay inside the tube. To be more precise, we proceed as follows. Fix a positive number p (0, 1). Let c(z) be the subset of Γ z with smallest volume such that e βv dσ(x) = p e βv dσ(x), (9) c(z) q(x)=z The union of the c(z) for z [0, 1] defines the effective transition tube. The above discussion establishes a most remarkable fact, namely if considered in configuration space, the ensemble of reactive trajectories can be characterized by equilibrium distributions on the iso-committor surfaces. This is the basic idea behind the finite-temperature string method. It is easy to see that if the potential is smooth, then in the zero-temperature limit, the transition path ensemble or the transition tube reduces to the minimum energy path. 9

10 III. FINITE-TEMPERATURE STRING METHOD A. Derivation For numerical purpose we will label the iso-committor surfaces by α [0, 1] which is an increasing function of z, e.g. the normalized arc-length of the centerline of the transition tube. We make the following two assumptions: 1. The transition tube is thin compared to the local radius of curvature of the centerline. 2. The iso-committor surfaces can be approximated by hyperplanes within the transition tube. The precise form of the second assumption is given in (30). By this assumption we avoid the intersection of the hyperplanes inside the transition tube. Our objective is to find the transition tube as well as the reaction coordinate q(x) whose level sets are the iso-committor surfaces. For this purpose it is useful to recall a variational formulation of q(x): As the solution of (4), q(x) has an equivalent characterization as being the minimizer of I = Ω\(A B) subject to the boundary conditions in (4). e βv q 2 dx. (10) In general this variational problem is difficult to solve since it is on a large dimensional space. But under the assumptions above, we can make an approximation by restricting the trial functions to functions whose level sets are hyperplanes. We will represent the oneparameter family of hyperplanes, {P α, α [0, 1]}, by their unit normal ˆn(α) and the mean position in the plane with respect to the equilibrium distribution restricted to P α, i.e. P ϕ(α) = x Pα α xe βv dσ(x) P α e βv dσ(x). (11) It was shown in Ref. 8 and 9 (see also the Appendix) that (10) reduces to I = 1 0 (f (α)) 2 e βf (α) ˆn ϕ 1 dα (12) under the assumptions described above. Here f(α) = q(ϕ(α)), the prime denotes the derivative with respect to α, and F (α) is the free energy F (α) = β 1 log e βv (x) dσ(x). (13) P α 10

11 The minimization condition for this functional is given by: ˆn(α) ϕ (α) (14) where ϕ (α) is the tangent vector of the string at parameter α, and α 0 f(α) = eβf (α ) dα 1 0 eβf (α ) dα. (15) Equation (14) says that the centerline ϕ(α) must be normal to the family of hyperplanes. A simple application of Laplace s method on the integrals in (15) implies that the isocommittor surface at α where f(α ) = 1 2 roughly coincides with the surface on which F attains maximum. Therefore the transition state ensemble can be characterized by either of the following: (i). F (α) is within k B T of its maximum value; (ii). f(α) 1 2. We emphasize that for (i) to be true, the correct free energy has to be used, i.e. one has to select the right reaction coordinate. We also note that the transition state region can be quite broad, if F or f changes slowly at α. A rigorous derivation of the conditions (14) and (15) can be found in Ref. 8. Heuristically, the minimum of the factor ˆn ϕ 1 in (12) is achieved under the condition (14) (we assume the length of ϕ (α) is a constant). Then the functional (12) reduces to I = 1 0 e βf (α) (f (α)) 2 dα (16) up to a constant. The minimizer of (16) is given by (15), which is also the solution to the one-dimensional version of the backward Kolmogorov equation: with boundary conditions f(0) = 0 and f(1) = 1. F (α)f (α) + β 1 f (α) = 0 (17) B. Implementation The finite-temperature string method is an iterative method for solving ϕ(α) = x Pα, (18) 11

12 and ˆn(α) ϕ (α). (19) This is done by seamlessly combining sampling on the hyperplanes with the dynamics of the hyperplanes. In (18) and (19), one of the equations can be viewed as a kinematic constraint and the other as the equilibrium condition. In the discussions above, we parametrized the planes according to (18), then (19) came out as the equilibrium condition. In the implementation of FTS, particularly for systems with rough energy landscapes, we find it more convenient to parametrize the planes according to (19) and set up an iterative procedure to solve (18) in order to find the transition tube. The object that we iterate upon is the so-called string, which is normal to the hyperplanes. At steady state, it coincides with the centerline of the transition tube. At each iteration step n, we denote the mean string and associated perpendicular hyperplanes by ϕ n (α) and Pα n respectively, where α [0, 1] is a parameterization of the string. The next iteration is given by: ϕ n+1 (α ) = x P n α (20) where α = g(α) is obtained by reparameterization in order to enforce a particular constraint on the parametrization (say, by normalized arc-length). However, due to the roughness of the energy landscape, using (20) directly may lead to an unstable numerical algorithm. This difficulty is overcome by adding a smoothing step for the mean string. The computation of ϕ n+1 therefore follows four steps: a. Calculate the mean position ϕ(α) on Pα n by constrained dynamics; b. Smoothing ϕ(α) gives ϕ(α); c. Reparameterization of ϕ(α) gives ϕ n+1 (α); d. Reinitialization of the sampling on the new hyperplanes. Notice that the smoothing step is used to accelerate convergence and avoid numerical instability. The details of each step are given below. A schematic of the computation procedure is shown in figure 2. 12

13 a. Sampling the energy landscape around ϕ n. The computation of x P n α is done as follows. On each hyperplane Pα n a Langevin dynamics ẋ = ( V ) (x) + 2β 1 η (21) is carried out, where ( ) denotes the projection to the hyperplane Pα n, and η is a white-noise forcing. The mean position of the system on P n α is calculated from (21): ϕ(α) = 1 T 1 T 0 T1 T 0 x(t)dt (22) where T 0 is the relaxation time and T 1 is the total simulation time. Constrained molecular dynamics can be used as well to calculate the mean position. For systems with very rough energy landscapes, it is convenient to add a constraining potential that localizes the sampling in (21) to region close to the current mean string. This is again used to avoid numerical instability, but has no effect on the final result. The following potential is used in our calculation 0, r < c V c (x) = (r d) 12 2((c d)(r d)) 6, c r < d, r d. Here r = x ϕ n, ϕ n is the current mean string and c and d are two parameters. Under this constraining force, sampling in (21) is confined in a tube with radius d around the current mean string. Moreover, the system experiences no constraining force in a even smaller tube with radius c. The constraining force is gradually switched off as the computation converges. This is done by gradually increase the confining parameter c. b. Smoothing the mean string. The calculated mean string ϕ(α) may not be very smooth and this may make the tangent vector so inaccurate that it introduces numerical instabilities. Therefore at each iteration, we smooth out ϕ(α) by solving the following optimization problem min ϕ(α) 1 0 (23) (C 1 κ(α) + C 2 ϕ(α) ϕ(α) M ) dα. (24) Here κ(α) is the curvature of ϕ(α), x M = (x, Mx) 1/2 is the L 2 norm of x weighted by M = (C(α) + I) 1 where C(α) is the covariance matrix of x generated by constrained 13

14 dynamics on each hyperplane. The parameters C 1, C 2 [0, 1] determine the relative weights of the two terms. As two extreme cases, the solution of (24) is a line segment when C 1 = 1, C 2 = 0 and the solution ϕ(α) = ϕ(α) when C 1 = 0, C 2 = 1. The minimization problem (24) can be solved by the steepest descent method, or more advanced techniques, e.g. the quasi-newton (BFGS) method. c. Reparameterization. At the third step of each iteration, the mean string ϕ(α) is reparameterized, for example, by normalized arc-length, to give ϕ n+1 (α). The reparameterization is done by polynomial interpolation. 1 d. Reinitialization. A new collection of hyperplanes P n+1 α which are orthogonal to ϕ n+1 (α) are obtained after the reparameterization step. To start the sampling process on the new planes, we need to generate the initial configurations for the Langevin dynamics (21). A naive way to generate the initial configurations is to project the realizations on P n α n+1 from the previous sampling process onto the new planes Pα. But unfortunately this procedure usually result in abrupt changes in the molecular structure and thus very large potential force. In the case that the artificial constraining potential (23) is used, the direct projection becomes even worse since it is possible that the projected realizations are out of the tube defined by the constraining potential and thus have infinite force. In our implementation, this difficulty is overcome by inserting a few intermediate hyperplanes, denoted by { P j, j = 0, 1,, J}, in between P n α n+1 and Pα, and each time we project the realization from P j to P j+1, followed by a few steps of relaxation on P j+1. This is analogous to choosing small time steps for numerical stability when solving ODEs or PDEs. Here P 0 = Pα n, P J = Pα n+1, and { P j, j = 1,, J 1} are obtained by linear interpolation between P n α and P n+1 α, i.e. the unit normals are given by ((1 jj )ˆn0 + jj ) ˆnJ, j = 1, 2, J 1 (25) ˆn j = c where ˆn 0 and ˆn J are the unit normals to P n α and P n+1 α factor. The interpolated plane P j goes through φ j where φ j is given by respectively, and c is the normalization φ j = (1 j J )φ0 + j J φj, j = 1, 2, J 1 (26) where φ 0 = ϕ n α and φj = ϕ n+1 α. On each interpolated hyperplane P j, the constraining force (23) is centered at φ j. 14

15 For each α, we move the realization x on P n α j = 1, 2,, J, we first project x from P j 1 to P j : to P n+1 α in J steps. At the j-th step, x := x ((x φ j ) ˆn j )ˆn j. (27) Then we relax the projected configuration on P j for a short time T 2 (a few time steps): ẋ(t) = ( V ),j (x) + 2β 1 η,j, 0 t T 2 (28) where ( V ),j = V ( V, ˆn j )ˆn j and V (x) includes the potential of the system and the constraining potential. Similar projection is done for η. In the following application to the alanine dipeptide, the parameter T 2 is taken to be a few time steps for (28), and J = 10. Sampling on the new hyperplanes P n+1 begins after we obtain the new set of realizations, and the above four-step procedure repeats. When the iteration converges with satisfactory accuracy, the mean string gives the centerline of the tube, the variance of x(α) on the hyperplanes give the width of the transition tube, and the function f(α) in (15) gives the committor value of each plane. To obtain f(α), the free energy (13) must be computed, which can be done by calculating the mean force by sampling via (21) on the final planes, and thermodynamic integration. IV. APPLICATION TO THE ALANINE DIPEPTIDE The ball and stick model of the alanine dipeptide is shown in figure 3. Despite its simple chemical structure, it exhibits some of the important features common to biomolecules, with the backbone degree of freedoms (dihedral angles φ and ψ), three methyl groups (CH 3 ), as well as the polar groups (N-H, C=O) which form hydrogen bonds with water in aqueous solution. The isomerization of alanine dipeptide has been the subject of several theoretical and computational studies and therefore serves as an excellent test for FTS. Apostolakis et al calculated the transition pathways and barriers for the isomerization of alanine dipeptide. 13 In their work, a two-dimensional potential of mean force on the φ- ψ plane was first calculated by the adaptive umbrella sampling scheme. The conjugate peak refinement algorithm and targeted molecular dynamics technique were applied on the 15

16 obtained two-dimensional potential to obtain the transition pathways. As a result, the mechanism of the transition is pre-assumed to depend only on the two torsion angles φ and ψ. Bolhuis et al applied TPS to study the isomerization of alanine dipeptide in vacuum and in solution. 14 Their calculation used all-atom representation for the molecule and explicit solvent model. An ensemble of transition pathways was collected, from which the transition state ensemble was found by determining the committor for each configurations visited by the trajectories. Their analysis shows that more degrees of freedom than the two torsion angle are necessary to describe the reaction coordinates. Recently, Ma and Dinner introduced a statistical method for identifying reaction coordinates from a database of candidate physical variables, and applied their method to study the isomerization of alanine dipeptide. 11 In the following, we apply FTS to study the transition of the alanine dipeptide both in vacuum and in explicit solvent. A. Alanine dipeptide in vacuum We use the full atomic representation of the alanine dipeptide molecule with the CHARMM22 force field. 15 We have not used the CMAP correction to the dihedral angle potentials 17 since it is unclear how the CMAP correction influences the dynamics of proteins. Figure 4 shows the adiabatic energy landscape of the molecule as a function of the two backbone dihedral angles φ and ψ. There are two stable conformers C 7eq and C ax. The state C 7eq is further split into two sub-states (denoted by C 7eq and C 7eq in figure 4) separated by a small barrier. Normally, one needs to identify the initial and final states before FTS can be used. This case is different since there is some periodicity in the system. The initial string (and the realizations on each hyperplane) is obtained by rotating the two backbone dihedral angles of the dipeptide along the diagonal line from ( 180, 180 ) to (180, 180 ), with all other internal degrees of freedom kept fixed. Only non-hydrogen atoms are represented on the string. The string is discretized using 40 points and correspondingly 40 hyperplanes are evolved in the computation. At each iteration, time averaging is performed for 0.5ps following the dynamics described in (21) with T = 272K. A hard-core constraining potential is added to restrict the sampling to a tube with width 0.2Å (c = 0.2 and d = 1 in (23)) around the string. This constraint is gradually removed as the string converges to the steady 16

17 state. The parameters used in smoothing the string are C 1 = C 2 = 1. To fix the overall rotation and translation of the molecule, we constrained one Carbon atom at the origin, one neighboring Carbon atom at the positive x-axis, and one Nitrogen atom in the xy-plane by harmonic springs. The computation converges after about iterations. It takes about several-hour CPU time on a single processor PC. The converged string and the tube are shown in figure 5. This tube is represented by projecting onto the φ-ψ plane the ensemble of the realizations on each hyperplane corresponding to the converged string. As we discussed earlier, this tube should coincide with the tube obtained from the most probable transition paths between the stable conformers C ax and C 7eq. The free energy F along the string is shown in figure 6 (upper panel). The free energy has three local maxima S 1, S 2 and S 3, corresponding to two transition pathways from C ax to C 7eq : (a) C ax - S 1 - C 7eq - S 2 - C 7eq ; and (b) C ax - S 3 - C 7eq. The transition at S 1 and S 3 is quite sharp. The energy difference of C ax relative to C 7eq is about E = 2.5 kcal/mol. The energy barrier is about E = 7.0 kcal/mol at S 1, and about E = 7.5 kcal/mol at S 3. Path (a) goes through an intermediate metastable state C 7eq around ( 150, 170 ). But the energy barrier from C 7eq to C 7eq is comparable to k B T and it has little effect on the transition from C ax to C 7eq. Shown in the lower panel of figure 6 is the committor function f(α) to the state C ax for the pathway C 7eq - S 3 - C ax calculated using the formula (15). The transition state region is identified as the hyperplane labeled by α that satisfies f(α ) 1. This plane is close to 2 the one with maximum free energy between C 7eq and C ax Figure 7 shows the projection of the level curves of the equilibrium probability density on the hyperplanes corresponding to the local maxima of the free energy. We see that at S 1 and S 2, the projected transition states are localized on lines which are roughly perpendicular to the projected mean path (dotted line), which suggests that the two backbone dihedral angles may parametrize reasonably well the reaction coordinate q(x) for this transition. However, the projected transition states at S 3 spread out in φ-ψ plane and are no longer concentrated on the line perpendicular to the projected mean path. This indicates that in addition to φ and ψ, there are other degree of freedoms that are important for this transition. In Ref. 14, the calculation by TPS shows that an additional torsion angle θ plays an important role in this transition, and the angles φ and θ provides a good set of coarse variables to parametrize the reaction coordinate q(x). However, our calculation shows that the angle 17

18 θ is approximately constant along the transition tube, and the projected transition states on φ-θ plane is still quite broad and far from being concentrated on the line perpendicular to the projected mean string (see figure 8). Therefore, we conclude that φ and θ are not sufficient to parametrize the reaction coordinate q(x). This discrepancy may be due to the different force fields used in the two calculations: We used the CHARMM22 force field, and the authors in Ref. 14 used AMBER. To see that we indeed obtained the right transition tube and transition state ensemble, we computed the committor distribution of the plane P α for which f(α ) 1 2 by initiating trajectories from this plane (this plane corresponds to the transition state region S 3 ). Recall that in our calculation the string is discretized to a collection of points parametrized by {α i }. In general α in figure 6 is not in this discrete set. To compensate for this, we compute the committor distribution for the two neighboring hyperplanes, one on each side of α. We randomly picked 2490 points based on the equilibrium distribution on each of the two planes. Starting from each of these points, 5000 trajectories were launched, from which the committor distribution P Cax or the probability that the trajectory goes to C ax is calculated. Figure 9 shows the distribution of P Cax. The distribution is peaked at P Cax = 0.3 and P Cax = 0.7 respectively. This confirms our prediction that the transition state region must be in between these two hyperplanes. We next refined the string and added one more plane between the two neighboring planes discussed above. The committor distribution of the new hyperplane is shown in figure 10. The peak is located around 1 2 indicating that we indeed obtained the right transition state ensemble. The small deviation of the peak from 1 2 is due to the approximation to the transition state surface by hyperplanes, and also to sampling errors in estimating the committor values. In figure 11, we plot four of the trajectories initiated from S 1. They stay well within the transition tube that we obtained by FTS. It is interesting to note that computing the committor distribution of P α to confirm the result of FTS is much more expensive than obtaining P α by FTS. B. Alanine dipeptide in explicit solvent The system consists of a full atom representation of an alanine dipeptide molecule modeled with the CHARMM22 force field, and 198 water molecules in a periodic cubic box. 18

19 The temperature of the solvated system is at K. The side length of the cubic box is 18.4Å. Ewald summation was used to treat the long-range Coulomb force. Only the backbone atoms on the dipeptide are represented on the string. Other atoms on the dipeptide as well as the water molecules are not included in the string, but they contribute to the force field as well as the width of the transition tube. For the initial string, we rotated ψ from 180 to 180 while keeping φ fixed at 50. The string is discretized using 30 points in the configuration space, and correspondingly 30 hyperplanes are evolved. To remove the rigid body translation and rotations for the dipeptide, we used a frame of reference which moves with the dipeptide. At each iteration, a constrained Langevin dynamics is carried out for 2 ps on each hyperplane in order to calculate the new position of the string. A hard core constraining potential is applied to restrict the Langevin dynamics to a neighborhood of width 0.4Å (c = 0.4 and d = 1 in (23)) of the string. The parameters used in smoothing the string are C 1 = C 2 = 1. The computation converged to a steady state at 45 iterations of 40,000 dynamics steps. The mean string was subsequently fixed, the constraining potential was removed, and the distributions in the hyperplanes were sampled. Figure 12 shows the converged transition tube on the φ-ψ plane. The background of the figure shows iso-contours of the free energy as a function of the φ-ψ variables. The free energy was obtained from 50 adaptive umbrella sampling 22 simulations of 100 ps at 300 K using the CHARMM program. 16 The biasing potential of those simulations was described by a combination of 12 harmonic functions and a constant in each dimension. Compared to the alanine dipeptide in vacuum, the angle ψ of C 7eq is shifted to a higher value by about 90. In addition, the helical metastable state has appeared. This state has a significant π-helix contribution in the CHARMM22 potential without the CMAP correction. 17 There are two transition path ensembles from C 7eq to α R : (a) C 7eq - S 1 - α R ; and (b) C 7eq - S 2 - α R. The projected hyperplanes onto the φ ψ plane concentrate on the lines perpendicular to the projected string. Therefore, the two backbone dihedral angles, φ and ψ, are qualified as a set of coarse variables to parametrize the reaction coordinate. Note that the transition tube is curved and deviates from vertical lines, therefore both φ and ψ are necessary to describe the transition dynamics, but they may not be sufficient. This is consistent with the conclusion in Ref. 14, where Bolhuis and coauthors calculated the committor distribution for configurations with ψ = 60. A uniform distribution was obtained from which the author 19

20 inferred that the reaction coordinate includes other important degrees of freedom, e.g. the solvent motion. To see whether the computed transition tube identifies with reasonable accuracy the transition state region, we computed the committor distribution on the hyperplane where the committor estimate was closest to 1/2. This is the plane centered around ( 96, 51.5 ) in the φ-ψ map. We initiated 100 trajectories from each of 896 points sampled according to the equilibrium distribution restricted to the hyperplane. We used finite-friction Langevin dynamics with a friction coefficient of 20 ps 1 as implemented in the CHARMM program. 16 The trajectory was stopped once the ψ angle was larger than 100 or smaller than 20 and the trajectory was counted as committed to the extended or the helical basin respectively. Due to the very long commitment times required by these trajectories, we limited ourselves to a smaller ensemble than in the vacuum study. The calculated committor distribution is shown in figure 13. The distribution is peaked around 1, but it is less peaked than the 2 one obtained in vacuum. The average value of the committor distribution is 0.50 and the standard deviation is One component of the distribution width is due to the limited sampling of the ensemble by only 100 trajectories per initial condition. The limited sampling by only 100 trajectories per initial condition contributes a standard deviation of 0.05 to the results of sampling a sharp 0.5 committor iso-surface. The remaining part of the error is due to the approximation of the hyper-surfaces by hyperplanes and due to the assumption that the isocommittors can be described on the peptide backbone degrees of freedom. Figure 14 shows four typical reactive trajectories. These trajectories have equal probability to fall into C 7eq or α R, and they follow the transition tube quite well. Compared to the behavior in vacuum, the dynamics of the solvated system is quite diffusive due to the collisions with water molecules and the lack of a high energetic barrier in the transition region. V. CONCLUSIONS In this paper FTS was successfully applied to study the transition events of the alanine dipeptide in vacuum and in explicit solvent. The results of FTS were justified by launching trajectories from the transition state region and the calculation of the committor distributions to confirm that FTS allows one indeed to find iso-committor surfaces, in particular the 20

21 one with committor value 1 corresponding to the transition state region. 2 It is illuminating to make a comparison between the philosophies behind TPS and FTS. TPS and FTS are after the same objects, namely the ensemble of transition paths and transition states. Their key difference is in the way that the transition paths are parameterized. TPS considers transition paths in the space of physical trajectories, parametrized by the real time. FTS considers transition paths in configuration space, parametrized, for example, by the arc-length of the centerline of the transition tube. This different parametrization gives FTS considerable advantage. First, the ensemble of transition paths can be characterized by equilibrium densities, if we consider where they hit the iso-committor surfaces. Secondly, this allows us to give a direct link between the transition path ensemble and the transition state ensemble. In addition, it also allows us to develop sampling procedures by which a collection of paths are harvested at the same time, instead of one by one as in TPS. In other words FTS allows one to bypass completely the calculation of reactive trajectories parametrized by time and directly obtain the iso-committor surfaces and the transition tubes which are more relevant to describe the mechanism of the transition. To conclude, FTS is a powerful tool for determining effective transition pathways in complex systems with multiscale energy landscapes. It does not require specifying a reaction coordinate beforehand. It allows us to determine the hyperplanes which are approximations to the iso-committor surfaces in configuration space by evolving a smooth curve. The smooth curve converges to the center of a tube by which transitions occur with high probability. Acknowledgments The authors thank M. Karplus for helpful discussions. E and Vanden-Eijnden was partially supported by ONR grants N and N , and NSF grants DMS , DMS and DMS PM acknowledges support by the Marie Curie European Fellowship grant number MEIF-CT The explicit water calculations were performed in the Crimson Grid cluster at Harvard, the SDSC teragrid and the CIMS computer center. 21

22 APPENDIX We derive the formula (12) from (10): I = q 2 e βv dx Ω 1 = q 2 e βv δ(q(x) z)dzdx Ω 0 1 = dz q 2 e βv δ(q(x) z)dx 0 Ω 1 = f (α)dα q e βv δ(ˆn (x ϕ(α)))dx 0 Ω 1 = f (α) q(ϕ) dα e βv δ(ˆn (x ϕ(α)))dx 0 Ω = 1 0 f (α) 2 (ˆn ϕ ) 1 e βf (α) dα. Here we denote Ω\(A B) by Ω. To go from the third to the fourth line, we made the assumption that the iso-committor surface is locally planar with normal ˆn, and defined f(α) = q(ϕ(α)). To go from the fourth to the fifth line, we used the following assumption (ˆn (x ϕ)) 2 Pα (ˆn ϕ ) 2 (30) where the average is with respect to the equilibrium distribution restricted to P α. In the last step, we defined the free energy F (α) as in (13) and used f (α) = q(ϕ) ϕ = q(ϕ) (ˆn ϕ ). (29) Electronic address: weiqing@math.pinceton.edu Electronic address: eve2@cims.nyu.edu Electronic address: plm@tammy.harvard.edu Electronic address: weinan@math.princeton.edu 1 W. E, W. Ren, and E. Vanden-Eijnden, Phys. Rev. B 66, (2002) 2 W. E, W. Ren, and E. Vanden-Eijnden, J. Phys. Chem. 109, 6688 (2005) 3 H. Jónsson, G. Mills, and K. W. Jacobsen, Nudged elastic band method for finding minimum energy paths of transitions. In: Classical and Quantum Dynamics in Condensed Phase Simulations. Edited by: Berne, B. J.; Ciccoti, G.; Coker, D. F. World Scientific, W. E, W. Ren, and E. Vanden-Eijnden, unpublished 22

23 5 E. A. Carter, G. Ciccotti, J. T. Hynes, and R. Kapral, Chem. Phys. Lett. 156, 472 (1989) 6 P. G. Bolhuis, D. Chandler, C. Dellago, and P. Geissler, Ann. Rev. Phys. Chem. 59, 291 (2002) 7 C. Dellago, P. G. Bolhuis, and P. L. Geissler, Adv. Chem. Phys. 123 (2002) 8 W. E and E. Vanden-Eijnden, Metastability, conformation dynamics, and transition pathways in complex system, in: Multiscale Modeling and Simulation, S. Attinger and P. Koumoutsakos eds. LNCSE 39, Springer, W. E, W. Ren, and E. Vanden-Eijnden, submitted to Chem. Phys. Lett. 10 R. Olender and R. Elber, J. Chem. Phys. 105, 9299 (1996) 11 A. Ma and A. Dinner, J. Phys. Chem. 109, 6769 (2005) 12 C. W. Gardiner, Handbook of stochastic methods for physics, chemistry, and the natural sciences. Springer, J. Apostolakis, P. Ferrara, and A. Caflisch, J. Chem. Phys. 10, 2099 (1999) 14 P. G. Bolhuis, C. Dellago, and D. Chandler, Proc. Natl. Acad. Sci. 97, 5877 (2000) 15 A. D. MacKerell, D. Bashford, M. Bellott, R. L. Dunbrack, J. D. Evanseck, M. J. Field, S. Fischer, J. Gao, H. Guo, S. Ha, D. Joseph-McCarthy, L. Kuchnir, K. Kuczera, F. T. K. Lau, C. Mattos, S. Michnick, T. Ngo, D. T. Nguyen, B. Prodhom, W. E. Reiher, B. Roux, M. Schlenkrich, J. C. Smith, R. Stote, J. Straub, M. Watanabe, J. Wiorkiewicz-Kuczera, D. Yin, and M. Karplus, J. Phys. Chem. B 102, (1998) 16 B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and M. Karplus, J. Comp. Chem. 4, (1983) 17 M. Feig, A. D. MacKerell, and C. L. Brooks, J. Phys. Chem. B 107, 2831 (2003) 18 S. Fischer, and M. Karplus, Chem. Phys. Lett. 194, (1992) 19 I. V. Ionova, and E. A. Carter, J. Chem. Phys. 98, (1993) 20 G. Henkelman, and H. Jónsson, J. Chem. Phys. 111, (1999) 21 L. R. Pratt, J. Chem. Phys. 85, (1986) 22 C. Bartels, and M. Karplus, J. Comput. Chem. 18, (1997) 23

24 LIST OF FIGURES FIG. 1: Transition tube between the metastable sets A and B. Also shown are the isocommittor surfaces Γ z. FIG. 2: Schematic of the computation procedure. Top figure (1st step): Sampling the hyperplane Pα n. The empty circles are the mean positions on the hyperplanes; Second figure (2nd step): Smoothing ϕ(α) to obtain ϕ(α); Third figure (3rd step): Reparameterization of ϕ(α) to obtain ϕ n+1 (α); Last figure (4th step): Reinitializing the sampling process on P n+1. The realizations on P n (empty circles) are moved to P n+1 (empty diamonds). FIG. 3: Schematic representation of the alanine dipeptide (CH 3 -CONH-CHCH 3 -CONH-CH 3 ). The backbone dihedral angles are labeled by φ: C-N-C-C and ψ: N-C-C-N. The picture is taken from Ref. 14. FIG. 4: Adiabatic energy landscape of the alanine dipeptide. The energy landscape is obtained by minimizing the potential energy of the molecule with (φ, ψ) fixed. The contours are drawn at multiples of 1 kcal/mol above the C 7eq minimum. There are two stable conformers C 7eq and C ax. FIG. 5: The transition tube between C 7eq and C ax. Two pathways exist: (a)c ax - S 1 - C 7eq - S 2 - C 7eq ; and (b): C ax - S 3 - C 7eq. The dotted line is the path of (φ, ψ) along the mean string. The converged hyperplanes across the points denoted by diamonds are approximations to the transition state surfaces. FIG. 6: Top figure: Free energy of the alanine dipeptide along the transition tube shown in figure 5; Lower figure: Committor f(α) along the path C 7eq - S 3 - C ax. The transition state ensemble for the transition C 7eq - S 3 - C ax is on the hyperplane labeled by α determined by f(α ) = 1 2. Note that this plane is also very close to the one with maximum free energy between C 7eq and C ax. FIG. 7: Transition state ensemble projected onto φ-ψ plane. The level curves show the density of the realizations on the hyperplanes corresponding to the local maxima of the free energy. These planes have committor value f(α) very close 1 2 for the corresponding transition. 24

25 FIG. 8: Transition tube projected onto φ-θ plane. The level curves show the density of the transition states at S 3. FIG. 9: Committor distribution for the transition state region S 3. Two histograms correspond to the two hyperplanes with index α directly above and below α, where f(α ) = 1. 2 FIG. 10: Committor distribution for the refined hyperplane with committor value 1. 2 FIG. 11: Four trajectories initiated from the transition state region S 1. The trajectories have equal probabilities to go to C 7eq and C ax. The trajectories stay in the transition tube obtained by string method. FIG. 12: Transition tube between C 7eq and α R projected onto the φ-ψ plane. The contour plot shows the potential of mean force of the φ-ψ angles. The iso-contours are separated by 0.5 kcal/mol. The thick black points that are connected with lines show the location of the mean string. FIG. 13: Committor distribution function for the transition state region S 1 which is the plane centered around ( 96, 51.5 ) in the φ-ψ map. FIG. 14: Four typical trajectories initiated from the transition state region S 1. The trajectories have equal probability to go to C 7eq and α R, and they follow the transition tube. 25

26 Γ z A B FIG. 1: 26

27 ~ ϕ A ϕ n B A ϕ _ ϕ n B ϕ n+1 A ϕ n B ϕ n+1 A ϕ n B FIG. 2: 27

28 FIG. 3: 28

29 180 C 7eq 90 C 7eq ψ 0 C ax φ FIG. 4: 29

30 FIG. 5: 30

31 8 7 S 3 S 1 6 E (kcal/mol) C ax 1 0 S 2 C 7eq C 7eq C 7eq α f(α) α* 0.4 α FIG. 6: 31

32 180 C 7eq S 2 90 C 7eq ψ 0 S 3 90 C ax S φ FIG. 7: 32

33 FIG. 8: 33

34 4 3 P(p Cax ) p Cax 4 3 P(p Cax ) p Cax FIG. 9: 34

35 3 2 P(p Cax ) p Cax FIG. 10: 35

36 FIG. 11: 36

37 FIG. 12: 37

Transition pathways in complex systems: Application of the finite-temperature string method to the alanine dipeptide

Transition pathways in complex systems: Application of the finite-temperature string method to the alanine dipeptide THE JOURNAL OF CHEMICAL PHYSICS 23, 3409 2005 Transition pathways in complex systems: Application of the finite-temperature string method to the alanine dipeptide Weiqing Ren a Department of Mathematics,

More information

Transition pathways in complex systems: Reaction coordinates, isocommittor surfaces, and transition tubes

Transition pathways in complex systems: Reaction coordinates, isocommittor surfaces, and transition tubes Chemical Physics Letters 413 (25) 242 247 www.elsevier.com/locate/cplett Transition pathways in complex systems: Reaction coordinates, isocommittor surfaces, and transition tubes Weinan E a, Weiqing Ren

More information

Improved Resolution of Tertiary Structure Elasticity in Muscle Protein

Improved Resolution of Tertiary Structure Elasticity in Muscle Protein Improved Resolution of Tertiary Structure Elasticity in Muscle Protein Jen Hsin and Klaus Schulten* Department of Physics and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois

More information

Molecular Dynamics Investigation of the ω-current in the Kv1.2 Voltage Sensor Domains

Molecular Dynamics Investigation of the ω-current in the Kv1.2 Voltage Sensor Domains Molecular Dynamics Investigation of the ω-current in the Kv1.2 Voltage Sensor Domains Fatemeh Khalili-Araghi, Emad Tajkhorshid, Benoît Roux, and Klaus Schulten Department of Physics, Department of Biochemistry,

More information

Towards a Theory of Transition Paths

Towards a Theory of Transition Paths Journal of Statistical Physics, Vol. 123, No. 3, May 2006 ( C 2006 ) DOI: 10.1007/s10955-005-9003-9 Towards a Theory of Transition Paths Weinan E 1 and Eric Vanden-Eijnden 2 Received July 19, 2005; accepted

More information

22 Path Optimisation Methods

22 Path Optimisation Methods 22 Path Optimisation Methods 204 22 Path Optimisation Methods Many interesting chemical and physical processes involve transitions from one state to another. Typical examples are migration paths for defects

More information

W tot (z) = W (z) k B T ln e i q iφ mp(z i )/k B T, [1]

W tot (z) = W (z) k B T ln e i q iφ mp(z i )/k B T, [1] Supporting Text Simulation Details. Simulations were carried out with the program charmm (1) using the PARAM27 force field with standard lipid (2), protein (3), and TIP3P water model (4) parameters. The

More information

TECHNIQUES TO LOCATE A TRANSITION STATE

TECHNIQUES TO LOCATE A TRANSITION STATE 32 III. TECHNIQUES TO LOCATE A TRANSITION STATE In addition to the location of minima, the calculation of transition states and reaction pathways is an interesting task for Quantum Chemistry (QC). The

More information

Adaptive algorithm for saddle point problem for Phase Field model

Adaptive algorithm for saddle point problem for Phase Field model Adaptive algorithm for saddle point problem for Phase Field model Jian Zhang Supercomputing Center, CNIC,CAS Collaborators: Qiang Du(PSU), Jingyan Zhang(PSU), Xiaoqiang Wang(FSU), Jiangwei Zhao(SCCAS),

More information

Dynamic force matching: Construction of dynamic coarse-grained models with realistic short time dynamics and accurate long time dynamics

Dynamic force matching: Construction of dynamic coarse-grained models with realistic short time dynamics and accurate long time dynamics for resubmission Dynamic force matching: Construction of dynamic coarse-grained models with realistic short time dynamics and accurate long time dynamics Aram Davtyan, 1 Gregory A. Voth, 1 2, a) and Hans

More information

Supplementary Figures:

Supplementary Figures: Supplementary Figures: Supplementary Figure 1: The two strings converge to two qualitatively different pathways. A) Models of active (red) and inactive (blue) states used as end points for the string calculations

More information

Approach to Thermal Equilibrium in Biomolecular

Approach to Thermal Equilibrium in Biomolecular Approach to Thermal Equilibrium in Biomolecular Simulation Eric Barth 1, Ben Leimkuhler 2, and Chris Sweet 2 1 Department of Mathematics Kalamazoo College Kalamazoo, Michigan, USA 49006 2 Centre for Mathematical

More information

A climbing string method for saddle point search

A climbing string method for saddle point search THE JOURNAL OF CHEMICAL PHYSICS 138,134105(2013) A climbing string method for saddle point search Weiqing Ren 1,a) and Eric Vanden-Eijnden 2,b) 1 Department of Mathematics, National University of Singapore,

More information

Problem Set Number 01, MIT (Winter-Spring 2018)

Problem Set Number 01, MIT (Winter-Spring 2018) Problem Set Number 01, 18.377 MIT (Winter-Spring 2018) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) February 28, 2018 Due Thursday, March 8, 2018. Turn it in (by 3PM) at the Math.

More information

Transition Path Theory

Transition Path Theory Transition Path Theory E. Vanden-Eijnden Courant Institute of Mathematical Sciences, New York University New York, NY 10012 eve2cims.nyu.edu Eric Vanden-Eijnden E. Vanden-Eijnden: Transition Path Theory,

More information

A growing string method for determining transition states: Comparison to the nudged elastic band and string methods

A growing string method for determining transition states: Comparison to the nudged elastic band and string methods JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 17 1 MAY 2004 A growing string method for determining transition states: Comparison to the nudged elastic band and string methods Baron Peters Department

More information

Supporting Information for. The Origin of Coupled Chloride and Proton Transport in a Cl /H + Antiporter

Supporting Information for. The Origin of Coupled Chloride and Proton Transport in a Cl /H + Antiporter Supporting Information for The Origin of Coupled Chloride and Proton Transport in a Cl /H + Antiporter Sangyun Lee, Heather B. Mayes, Jessica M. J. Swanson, and Gregory A. Voth * Department of Chemistry,

More information

Membrane insertion profiles of peptides probed by molecular dynamics simulations

Membrane insertion profiles of peptides probed by molecular dynamics simulations Membrane insertion profiles of peptides probed by molecular dynamics simulations In-Chul Yeh, * Mark A. Olson, # Michael S. Lee, *# and Anders Wallqvist * * Biotechnology High Performance Computing Software

More information

Calculation of point-to-point short time and rare. trajectories with boundary value formulation

Calculation of point-to-point short time and rare. trajectories with boundary value formulation Calculation of point-to-point short time and rare trajectories with boundary value formulation Dov Bai and Ron Elber Department of Computer Science, Upson Hall 413 Cornell University, Ithaca, NY 14853

More information

Computing on Virtual Slow Manifolds of Fast Stochastic Systems.

Computing on Virtual Slow Manifolds of Fast Stochastic Systems. European Society of Computational Methods in Sciences and Engineering (ESCMSE) Journal of Numerical Analysis, Industrial and Applied Mathematics (JNAIAM) vol. 1, no. 1, nnnn, pp. 1-3 ISSN 179 814 Computing

More information

Path Optimization with Application to Tunneling

Path Optimization with Application to Tunneling Path Optimization with Application to Tunneling Dóróthea M. Einarsdóttir, Andri Arnaldsson, Finnbogi Óskarsson, and Hannes Jónsson Science Institute, VR-III, University of Iceland, 107 Reykjavík, Iceland

More information

Exploring the energy landscape

Exploring the energy landscape Exploring the energy landscape ChE210D Today's lecture: what are general features of the potential energy surface and how can we locate and characterize minima on it Derivatives of the potential energy

More information

Molecular dynamics (MD) and the Monte Carlo simulation

Molecular dynamics (MD) and the Monte Carlo simulation Escaping free-energy minima Alessandro Laio and Michele Parrinello* Centro Svizzero di Calcolo Scientifico, Via Cantonale, CH-6928 Manno, Switzerland; and Department of Chemistry, Eidgenössische Technische

More information

Numerical methods in molecular dynamics and multiscale problems

Numerical methods in molecular dynamics and multiscale problems Numerical methods in molecular dynamics and multiscale problems Two examples T. Lelièvre CERMICS - Ecole des Ponts ParisTech & MicMac project-team - INRIA Horizon Maths December 2012 Introduction The aim

More information

Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points

Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 22 8 DECEMBER 2000 Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points Graeme Henkelman a) and

More information

Calculation of Classical Trajectories with Boundary Value Formulation

Calculation of Classical Trajectories with Boundary Value Formulation Calculation of Classical Trajectories with Boundary Value Formulation R. Elber Department of Computer Science, Cornell University, 413 Upson Hall, Ithaca NY 14853 ron@cs.cornell.edu Ron Elber Ron Elber:

More information

Obtaining reaction coordinates by likelihood maximization

Obtaining reaction coordinates by likelihood maximization THE JOURNAL OF CHEMICAL PHYSICS 125, 054108 2006 Obtaining reaction coordinates by likelihood maximization Baron Peters and Bernhardt L. Trout a Department of Chemical Engineering, Massachusetts Institute

More information

New Physical Principle for Monte-Carlo simulations

New Physical Principle for Monte-Carlo simulations EJTP 6, No. 21 (2009) 9 20 Electronic Journal of Theoretical Physics New Physical Principle for Monte-Carlo simulations Michail Zak Jet Propulsion Laboratory California Institute of Technology, Advance

More information

CS273: Algorithms for Structure Handout # 2 and Motion in Biology Stanford University Thursday, 1 April 2004

CS273: Algorithms for Structure Handout # 2 and Motion in Biology Stanford University Thursday, 1 April 2004 CS273: Algorithms for Structure Handout # 2 and Motion in Biology Stanford University Thursday, 1 April 2004 Lecture #2: 1 April 2004 Topics: Kinematics : Concepts and Results Kinematics of Ligands and

More information

Finding minimum energy paths using Gaussian process regression

Finding minimum energy paths using Gaussian process regression Finding minimum energy paths using Gaussian process regression Olli-Pekka Koistinen Helsinki Institute for Information Technology HIIT Department of Computer Science, Aalto University Department of Applied

More information

1. Introductory Examples

1. Introductory Examples 1. Introductory Examples We introduce the concept of the deterministic and stochastic simulation methods. Two problems are provided to explain the methods: the percolation problem, providing an example

More information

Distance Constraint Model; Donald J. Jacobs, University of North Carolina at Charlotte Page 1 of 11

Distance Constraint Model; Donald J. Jacobs, University of North Carolina at Charlotte Page 1 of 11 Distance Constraint Model; Donald J. Jacobs, University of North Carolina at Charlotte Page 1 of 11 Taking the advice of Lord Kelvin, the Father of Thermodynamics, I describe the protein molecule and other

More information

Midterm 1 Review. Distance = (x 1 x 0 ) 2 + (y 1 y 0 ) 2.

Midterm 1 Review. Distance = (x 1 x 0 ) 2 + (y 1 y 0 ) 2. Midterm 1 Review Comments about the midterm The midterm will consist of five questions and will test on material from the first seven lectures the material given below. No calculus either single variable

More information

Hierarchical Metastable States and Kinetic Transition Networks: Trajectory Mapping and Clustering

Hierarchical Metastable States and Kinetic Transition Networks: Trajectory Mapping and Clustering Hierarchical Metastable States and Kinetic Transition Networks: Trajectory Mapping and Clustering Xin Zhou xzhou@gucas.ac.cn Graduate University of Chinese Academy of Sciences, Beijing 2012.6.5 KITP, Santa

More information

Title Theory of solutions in the energy r of the molecular flexibility Author(s) Matubayasi, N; Nakahara, M Citation JOURNAL OF CHEMICAL PHYSICS (2003), 9702 Issue Date 2003-11-08 URL http://hdl.handle.net/2433/50354

More information

CS 273 Prof. Serafim Batzoglou Prof. Jean-Claude Latombe Spring Lecture 12 : Energy maintenance (1) Lecturer: Prof. J.C.

CS 273 Prof. Serafim Batzoglou Prof. Jean-Claude Latombe Spring Lecture 12 : Energy maintenance (1) Lecturer: Prof. J.C. CS 273 Prof. Serafim Batzoglou Prof. Jean-Claude Latombe Spring 2006 Lecture 12 : Energy maintenance (1) Lecturer: Prof. J.C. Latombe Scribe: Neda Nategh How do you update the energy function during the

More information

1 Lyapunov theory of stability

1 Lyapunov theory of stability M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability

More information

Kinetic Pathways of Ion Pair Dissociation in Water

Kinetic Pathways of Ion Pair Dissociation in Water 3706 J. Phys. Chem. B 1999, 103, 3706-3710 Kinetic Pathways of Ion Pair Dissociation in Water Phillip L. Geissler, Christoph Dellago, and David Chandler* Department of Chemistry, UniVersity of California,

More information

All-atom Molecular Mechanics. Trent E. Balius AMS 535 / CHE /27/2010

All-atom Molecular Mechanics. Trent E. Balius AMS 535 / CHE /27/2010 All-atom Molecular Mechanics Trent E. Balius AMS 535 / CHE 535 09/27/2010 Outline Molecular models Molecular mechanics Force Fields Potential energy function functional form parameters and parameterization

More information

Vibrational Energy Relaxation of Tailored Hemes in Myoglobin Following Ligand Photolysis Supports Energy Funneling Mechanism of Heme Cooling

Vibrational Energy Relaxation of Tailored Hemes in Myoglobin Following Ligand Photolysis Supports Energy Funneling Mechanism of Heme Cooling 10634 J. Phys. Chem. B 2003, 107, 10634-10639 Vibrational Energy Relaxation of Tailored Hemes in Myoglobin Following Ligand Photolysis Supports Energy Funneling Mechanism of Heme Cooling Lintao Bu and

More information

Free energy calculations and the potential of mean force

Free energy calculations and the potential of mean force Free energy calculations and the potential of mean force IMA Workshop on Classical and Quantum Approaches in Molecular Modeling Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science

More information

Lecture 11: Potential Energy Functions

Lecture 11: Potential Energy Functions Lecture 11: Potential Energy Functions Dr. Ronald M. Levy ronlevy@temple.edu Originally contributed by Lauren Wickstrom (2011) Microscopic/Macroscopic Connection The connection between microscopic interactions

More information

Nested Stochastic Simulation Algorithm for Chemical Kinetic Systems with Disparate Rates. Abstract

Nested Stochastic Simulation Algorithm for Chemical Kinetic Systems with Disparate Rates. Abstract Nested Stochastic Simulation Algorithm for Chemical Kinetic Systems with Disparate Rates Weinan E Department of Mathematics and PACM, Princeton University, Princeton, NJ 08544, USA Di Liu and Eric Vanden-Eijnden

More information

Assignment 2: Conformation Searching (50 points)

Assignment 2: Conformation Searching (50 points) Chemistry 380.37 Fall 2015 Dr. Jean M. Standard September 16, 2015 Assignment 2: Conformation Searching (50 points) In this assignment, you will use the Spartan software package to investigate some conformation

More information

Reaction Pathways of Metastable Markov Chains - LJ clusters Reorganization

Reaction Pathways of Metastable Markov Chains - LJ clusters Reorganization Reaction Pathways of Metastable Markov Chains - LJ clusters Reorganization Eric Vanden-Eijnden Courant Institute Spectral approach to metastability - sets, currents, pathways; Transition path theory or

More information

Multi-Ensemble Markov Models and TRAM. Fabian Paul 21-Feb-2018

Multi-Ensemble Markov Models and TRAM. Fabian Paul 21-Feb-2018 Multi-Ensemble Markov Models and TRAM Fabian Paul 21-Feb-2018 Outline Free energies Simulation types Boltzmann reweighting Umbrella sampling multi-temperature simulation accelerated MD Analysis methods

More information

Peptide folding in non-aqueous environments investigated with molecular dynamics simulations Soto Becerra, Patricia

Peptide folding in non-aqueous environments investigated with molecular dynamics simulations Soto Becerra, Patricia University of Groningen Peptide folding in non-aqueous environments investigated with molecular dynamics simulations Soto Becerra, Patricia IMPORTANT NOTE: You are advised to consult the publisher's version

More information

Physics 202 Laboratory 5. Linear Algebra 1. Laboratory 5. Physics 202 Laboratory

Physics 202 Laboratory 5. Linear Algebra 1. Laboratory 5. Physics 202 Laboratory Physics 202 Laboratory 5 Linear Algebra Laboratory 5 Physics 202 Laboratory We close our whirlwind tour of numerical methods by advertising some elements of (numerical) linear algebra. There are three

More information

Analysis of the Lennard-Jones-38 stochastic network

Analysis of the Lennard-Jones-38 stochastic network Analysis of the Lennard-Jones-38 stochastic network Maria Cameron Joint work with E. Vanden-Eijnden Lennard-Jones clusters Pair potential: V(r) = 4e(r -12 - r -6 ) 1. Wales, D. J., Energy landscapes: calculating

More information

Effective dynamics for the (overdamped) Langevin equation

Effective dynamics for the (overdamped) Langevin equation Effective dynamics for the (overdamped) Langevin equation Frédéric Legoll ENPC and INRIA joint work with T. Lelièvre (ENPC and INRIA) Enumath conference, MS Numerical methods for molecular dynamics EnuMath

More information

Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games

Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Alberto Bressan ) and Khai T. Nguyen ) *) Department of Mathematics, Penn State University **) Department of Mathematics,

More information

Amyloid fibril polymorphism is under kinetic control. Supplementary Information

Amyloid fibril polymorphism is under kinetic control. Supplementary Information Amyloid fibril polymorphism is under kinetic control Supplementary Information Riccardo Pellarin Philipp Schütz Enrico Guarnera Amedeo Caflisch Department of Biochemistry, University of Zürich, Winterthurerstrasse

More information

Equilibrium Time Correlation Functions from Irreversible Transformations in Trajectory Space

Equilibrium Time Correlation Functions from Irreversible Transformations in Trajectory Space J. Phys. Chem. B 2004, 108, 6667-6672 6667 Equilibrium Time Correlation Functions from Irreversible Transformations in Trajectory Space Phillip L. Geissler*, Department of Chemistry, Massachusetts Institute

More information

Exercise 2: Solvating the Structure Before you continue, follow these steps: Setting up Periodic Boundary Conditions

Exercise 2: Solvating the Structure Before you continue, follow these steps: Setting up Periodic Boundary Conditions Exercise 2: Solvating the Structure HyperChem lets you place a molecular system in a periodic box of water molecules to simulate behavior in aqueous solution, as in a biological system. In this exercise,

More information

The Molecular Dynamics Method

The Molecular Dynamics Method The Molecular Dynamics Method Thermal motion of a lipid bilayer Water permeation through channels Selective sugar transport Potential Energy (hyper)surface What is Force? Energy U(x) F = d dx U(x) Conformation

More information

Computing Reaction Rates Using Macro-states

Computing Reaction Rates Using Macro-states Computing Reaction Rates Using Macro-states Badi Abdul-Wahid 1 Ronan Costaouec 2 Eric Darve 2 Michelle Feng 1 Jesús Izaguirre 1 1 Notre-Dame University 2 Stanford University Tuesday July 3rd 2012 14:30

More information

Gromacs Workshop Spring CSC

Gromacs Workshop Spring CSC Gromacs Workshop Spring 2007 @ CSC Erik Lindahl Center for Biomembrane Research Stockholm University, Sweden David van der Spoel Dept. Cell & Molecular Biology Uppsala University, Sweden Berk Hess Max-Planck-Institut

More information

Mechanics IV: Oscillations

Mechanics IV: Oscillations Mechanics IV: Oscillations Chapter 4 of Morin covers oscillations, including damped and driven oscillators in detail. Also see chapter 10 of Kleppner and Kolenkow. For more on normal modes, see any book

More information

Clusters and Percolation

Clusters and Percolation Chapter 6 Clusters and Percolation c 2012 by W. Klein, Harvey Gould, and Jan Tobochnik 5 November 2012 6.1 Introduction In this chapter we continue our investigation of nucleation near the spinodal. We

More information

Tangent spaces, normals and extrema

Tangent spaces, normals and extrema Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent

More information

Sample Exam 1 KEY NAME: 1. CS 557 Sample Exam 1 KEY. These are some sample problems taken from exams in previous years. roughly ten questions.

Sample Exam 1 KEY NAME: 1. CS 557 Sample Exam 1 KEY. These are some sample problems taken from exams in previous years. roughly ten questions. Sample Exam 1 KEY NAME: 1 CS 557 Sample Exam 1 KEY These are some sample problems taken from exams in previous years. roughly ten questions. Your exam will have 1. (0 points) Circle T or T T Any curve

More information

Molecular dynamics simulations of anti-aggregation effect of ibuprofen. Wenling E. Chang, Takako Takeda, E. Prabhu Raman, and Dmitri Klimov

Molecular dynamics simulations of anti-aggregation effect of ibuprofen. Wenling E. Chang, Takako Takeda, E. Prabhu Raman, and Dmitri Klimov Biophysical Journal, Volume 98 Supporting Material Molecular dynamics simulations of anti-aggregation effect of ibuprofen Wenling E. Chang, Takako Takeda, E. Prabhu Raman, and Dmitri Klimov Supplemental

More information

Lecture D4 - Intrinsic Coordinates

Lecture D4 - Intrinsic Coordinates J. Peraire 16.07 Dynamics Fall 2004 Version 1.1 Lecture D4 - Intrinsic Coordinates In lecture D2 we introduced the position, velocity and acceleration vectors and referred them to a fixed cartesian coordinate

More information

Free energy simulations

Free energy simulations Free energy simulations Marcus Elstner and Tomáš Kubař January 14, 2013 Motivation a physical quantity that is of most interest in chemistry? free energies Helmholtz F or Gibbs G holy grail of computational

More information

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 17 Feb 2007

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 17 Feb 2007 Solvent coarse-graining and the string method applied to the hydrophobic collapse of a hydrated chain arxiv:cond-mat/0702407v1 [cond-mat.stat-mech] 17 Feb 2007 Thomas F. Miller, III, Eric Vanden-Eijnden,

More information

Newtonian Mechanics. Chapter Classical space-time

Newtonian Mechanics. Chapter Classical space-time Chapter 1 Newtonian Mechanics In these notes classical mechanics will be viewed as a mathematical model for the description of physical systems consisting of a certain (generally finite) number of particles

More information

Importance Sampling in Monte Carlo Simulation of Rare Transition Events

Importance Sampling in Monte Carlo Simulation of Rare Transition Events Importance Sampling in Monte Carlo Simulation of Rare Transition Events Wei Cai Lecture 1. August 1, 25 1 Motivation: time scale limit and rare events Atomistic simulations such as Molecular Dynamics (MD)

More information

ICCP Project 2 - Advanced Monte Carlo Methods Choose one of the three options below

ICCP Project 2 - Advanced Monte Carlo Methods Choose one of the three options below ICCP Project 2 - Advanced Monte Carlo Methods Choose one of the three options below Introduction In statistical physics Monte Carlo methods are considered to have started in the Manhattan project (1940

More information

APPPHYS217 Tuesday 25 May 2010

APPPHYS217 Tuesday 25 May 2010 APPPHYS7 Tuesday 5 May Our aim today is to take a brief tour of some topics in nonlinear dynamics. Some good references include: [Perko] Lawrence Perko Differential Equations and Dynamical Systems (Springer-Verlag

More information

MD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky

MD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky MD Thermodynamics Lecture 1 3/6/18 1 Molecular dynamics The force depends on positions only (not velocities) Total energy is conserved (micro canonical evolution) Newton s equations of motion (second order

More information

Modeling the sputter deposition of thin film photovoltaics using long time scale dynamics techniques

Modeling the sputter deposition of thin film photovoltaics using long time scale dynamics techniques Loughborough University Institutional Repository Modeling the sputter deposition of thin film photovoltaics using long time scale dynamics techniques This item was submitted to Loughborough University's

More information

Supporting Online Material for

Supporting Online Material for www.sciencemag.org/cgi/content/full/309/5742/1868/dc1 Supporting Online Material for Toward High-Resolution de Novo Structure Prediction for Small Proteins Philip Bradley, Kira M. S. Misura, David Baker*

More information

Lecture 12: Detailed balance and Eigenfunction methods

Lecture 12: Detailed balance and Eigenfunction methods Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2015 Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility),

More information

Unconstrained optimization

Unconstrained optimization Chapter 4 Unconstrained optimization An unconstrained optimization problem takes the form min x Rnf(x) (4.1) for a target functional (also called objective function) f : R n R. In this chapter and throughout

More information

T6.2 Molecular Mechanics

T6.2 Molecular Mechanics T6.2 Molecular Mechanics We have seen that Benson group additivities are capable of giving heats of formation of molecules with accuracies comparable to those of the best ab initio procedures. However,

More information

Physics 106a, Caltech 4 December, Lecture 18: Examples on Rigid Body Dynamics. Rotating rectangle. Heavy symmetric top

Physics 106a, Caltech 4 December, Lecture 18: Examples on Rigid Body Dynamics. Rotating rectangle. Heavy symmetric top Physics 106a, Caltech 4 December, 2018 Lecture 18: Examples on Rigid Body Dynamics I go through a number of examples illustrating the methods of solving rigid body dynamics. In most cases, the problem

More information

Simulated Annealing for Constrained Global Optimization

Simulated Annealing for Constrained Global Optimization Monte Carlo Methods for Computation and Optimization Final Presentation Simulated Annealing for Constrained Global Optimization H. Edwin Romeijn & Robert L.Smith (1994) Presented by Ariel Schwartz Objective

More information

Linear & nonlinear classifiers

Linear & nonlinear classifiers Linear & nonlinear classifiers Machine Learning Hamid Beigy Sharif University of Technology Fall 1396 Hamid Beigy (Sharif University of Technology) Linear & nonlinear classifiers Fall 1396 1 / 44 Table

More information

Molecular Simulation II. Classical Mechanical Treatment

Molecular Simulation II. Classical Mechanical Treatment Molecular Simulation II Quantum Chemistry Classical Mechanics E = Ψ H Ψ ΨΨ U = E bond +E angle +E torsion +E non-bond Jeffry D. Madura Department of Chemistry & Biochemistry Center for Computational Sciences

More information

Hyeyoung Shin a, Tod A. Pascal ab, William A. Goddard III abc*, and Hyungjun Kim a* Korea

Hyeyoung Shin a, Tod A. Pascal ab, William A. Goddard III abc*, and Hyungjun Kim a* Korea The Scaled Effective Solvent Method for Predicting the Equilibrium Ensemble of Structures with Analysis of Thermodynamic Properties of Amorphous Polyethylene Glycol-Water Mixtures Hyeyoung Shin a, Tod

More information

Optimization-Based String Method for Finding Minimum Energy Path

Optimization-Based String Method for Finding Minimum Energy Path Commun. Comput. Phys. doi: 10.4208/cicp.220212.030812a Vol. 14, No. 2, pp. 265-275 August 2013 Optimization-Based String Method for Finding Minimum Energy Path Amit Samanta 1, and Weinan E 2,3 1 Program

More information

Predicting the Folding Pathway of Engrailed Homeodomain with a Probabilistic Roadmap Enhanced Reaction-Path Algorithm

Predicting the Folding Pathway of Engrailed Homeodomain with a Probabilistic Roadmap Enhanced Reaction-Path Algorithm 1622 Biophysical Journal Volume 94 March 2008 1622 1629 Predicting the Folding Pathway of Engrailed Homeodomain with a Probabilistic Roadmap Enhanced Reaction-Path Algorithm Da-wei Li,* Haijun Yang,* Li

More information

The dynamics of small particles whose size is roughly 1 µmt or. smaller, in a fluid at room temperature, is extremely erratic, and is

The dynamics of small particles whose size is roughly 1 µmt or. smaller, in a fluid at room temperature, is extremely erratic, and is 1 I. BROWNIAN MOTION The dynamics of small particles whose size is roughly 1 µmt or smaller, in a fluid at room temperature, is extremely erratic, and is called Brownian motion. The velocity of such particles

More information

Crystal Structure Prediction using CRYSTALG program

Crystal Structure Prediction using CRYSTALG program Crystal Structure Prediction using CRYSTALG program Yelena Arnautova Baker Laboratory of Chemistry and Chemical Biology, Cornell University Problem of crystal structure prediction: - theoretical importance

More information

Stochastic Histories. Chapter Introduction

Stochastic Histories. Chapter Introduction Chapter 8 Stochastic Histories 8.1 Introduction Despite the fact that classical mechanics employs deterministic dynamical laws, random dynamical processes often arise in classical physics, as well as in

More information

Supporting Information

Supporting Information Supporting Information Comparative Study of Materials-Binding Peptide Interactions with Gold and Silver Surfaces and Nanostructures: A Thermodynamic Basis for Biological Selectivity of Inorganic Materials

More information

Numerical Methods of Applied Mathematics -- II Spring 2009

Numerical Methods of Applied Mathematics -- II Spring 2009 MIT OpenCourseWare http://ocw.mit.edu 18.336 Numerical Methods of Applied Mathematics -- II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

More information

Effect of protein shape on multibody interactions between membrane inclusions

Effect of protein shape on multibody interactions between membrane inclusions PHYSICAL REVIEW E VOLUME 61, NUMBER 4 APRIL 000 Effect of protein shape on multibody interactions between membrane inclusions K. S. Kim, 1, * John Neu, and George Oster 3, 1 Department of Physics, Graduate

More information

On elastic lines of double curvature

On elastic lines of double curvature Sur les lignes élastiques à double coubure Correspondance sur l École Poly. 3 (1816) 355-360. On elastic lines of double curvature By Siméon Denis Poisson Translated by D. H. Delphenich In this article

More information

Lattice protein models

Lattice protein models Lattice protein models Marc R. Roussel epartment of Chemistry and Biochemistry University of Lethbridge March 5, 2009 1 Model and assumptions The ideas developed in the last few lectures can be applied

More information

Dynamics of Nucleation in the Ising Model

Dynamics of Nucleation in the Ising Model J. Phys. Chem. B 2004, 108, 19681-19686 19681 Dynamics of Nucleation in the Ising Model Albert C. Pan and David Chandler* Department of Chemistry, UniVersity of California, Berkeley, California 94720-1460

More information

Finite temperature analysis of stochastic networks

Finite temperature analysis of stochastic networks Finite temperature analysis of stochastic networks Maria Cameron University of Maryland USA Stochastic networks with detailed balance * The generator matrix L = ( L ij ), i, j S L ij 0, i, j S, i j j S

More information

27 : Distributed Monte Carlo Markov Chain. 1 Recap of MCMC and Naive Parallel Gibbs Sampling

27 : Distributed Monte Carlo Markov Chain. 1 Recap of MCMC and Naive Parallel Gibbs Sampling 10-708: Probabilistic Graphical Models 10-708, Spring 2014 27 : Distributed Monte Carlo Markov Chain Lecturer: Eric P. Xing Scribes: Pengtao Xie, Khoa Luu In this scribe, we are going to review the Parallel

More information

Computational Molecular Biophysics. Computational Biophysics, GRS Jülich SS 2013

Computational Molecular Biophysics. Computational Biophysics, GRS Jülich SS 2013 Computational Molecular Biophysics Computational Biophysics, GRS Jülich SS 2013 Computational Considerations Number of terms for different energy contributions / E el ~N ~N ~N ~N 2 Computational cost for

More information

Discriminative Direction for Kernel Classifiers

Discriminative Direction for Kernel Classifiers Discriminative Direction for Kernel Classifiers Polina Golland Artificial Intelligence Lab Massachusetts Institute of Technology Cambridge, MA 02139 polina@ai.mit.edu Abstract In many scientific and engineering

More information

Analysis of the simulation

Analysis of the simulation Analysis of the simulation Marcus Elstner and Tomáš Kubař January 7, 2014 Thermodynamic properties time averages of thermodynamic quantites correspond to ensemble averages (ergodic theorem) some quantities

More information

Equilibrium Structure and Folding of a Helix-Forming Peptide: Circular Dichroism Measurements and Replica-Exchange Molecular Dynamics Simulations

Equilibrium Structure and Folding of a Helix-Forming Peptide: Circular Dichroism Measurements and Replica-Exchange Molecular Dynamics Simulations 3786 Biophysical Journal Volume 87 December 2004 3786 3798 Equilibrium Structure and Folding of a Helix-Forming Peptide: Circular Dichroism Measurements and Replica-Exchange Molecular Dynamics Simulations

More information

Joint ICTP-IAEA Workshop on Physics of Radiation Effect and its Simulation for Non-Metallic Condensed Matter.

Joint ICTP-IAEA Workshop on Physics of Radiation Effect and its Simulation for Non-Metallic Condensed Matter. 2359-16 Joint ICTP-IAEA Workshop on Physics of Radiation Effect and its Simulation for Non-Metallic Condensed Matter 13-24 August 2012 Introduction to atomistic long time scale methods Roger Smith Loughborough

More information

Molecular Mechanics. I. Quantum mechanical treatment of molecular systems

Molecular Mechanics. I. Quantum mechanical treatment of molecular systems Molecular Mechanics I. Quantum mechanical treatment of molecular systems The first principle approach for describing the properties of molecules, including proteins, involves quantum mechanics. For example,

More information

THEODORE VORONOV DIFFERENTIAL GEOMETRY. Spring 2009

THEODORE VORONOV DIFFERENTIAL GEOMETRY. Spring 2009 [under construction] 8 Parallel transport 8.1 Equation of parallel transport Consider a vector bundle E B. We would like to compare vectors belonging to fibers over different points. Recall that this was

More information